I am not a native speaker, but according to my translation book, compromise translates to kompromiso.

Compromise means to make an agreement where one person/side gives up part of their demands.

So in your sentence the speaker tells the Church some information, but doesn't tell them the whole truth, because it's what the people of the Church wanted to hear. So the agreement was that the speaker will tell the Church information, and the demand he/she gave up was telling the Church the whole truth.

Compromise is a confusing word to understand. It essentially has two very similar meanings.

1. To reach and agreement/settlement that didn't fulfill all the original demands of one or more people/sides. Example: I wanted a blue car, but my wife wanted red. So we compromised on green which we both like. So the husband and wife both got a colour they like, but they didn't fulfill the original demand that was to get their favorite colour.They compromised on the colour of their car.

2. It can also mean to expose/jeopardise. Example: I compromised the houses security. So the house was exposed to dangers such as burglary and the safety of the occupants where jepodised.

"Compromised nutrition increases the risk of infection and prolongs recovery from acute illness."

In you newest sentence, compromised is used in the second meaning. So the health of the body was compromised i.e. it was exposed to infection, and the recovery from acute illness was jepodised, which was a result of inadequate/insufficient nutrition. But compromise doesn't literally mean inadequate/insufficient.